Question

A Ferris wheel has a radius of 73.7 meters, and (while spinning) one rotation takes 122 seconds.

Samuel gets into a carriage at the bottom of the Ferris wheel. At \(t=0\), the wheel starts spinning, and it spins continuously for 364 seconds until an abrupt stop. At that moment, Samuel drops his keys, which fall directly down to the ground.

How far, in meters, are the keys from where Samuel entered the carriage? The tolerance is \(\pm 0.1\) meters.

Solution

Since Samuel got on at the bottom of the wheel, he begins at the midline (horizontally). Thus, we can use a stretched, but unshifted, sine wave to model Samuel’s horizontal position over time.

\[x~=~A\cdot\sin(Bt)\]

The amplitude will match the radius of the wheel.

\[A=73.7\]

To get \(B\), the angular frequency, we divide \(2\pi\) by the period (which was given to us).

\[B~=~\frac{2\pi}{P}\]

\[B~=~\frac{2\pi}{122}\]

\[B~\approx~0.0515015\]

Notice,

\[x~=~73.7\cdot\sin(0.0515015\cdot 364)\]

\[x~=~-12.2087613\]

We only care about the absolute value of \(x\).

\[|x| ~=~ 12.2087613\]

Meta-information

extype: num exsolution: 12.2087613 exname: ferris_wheel_key_drop extol: 0.1